Deriving Differential Equations from the Riccati Equation for Optimal Control

[JavierOlivares]

Homework Statement

I was wondering if I can get some help on a Linear Regulator Problem for an Optimal Control Problem. Given a state equation and performance measure I am trying to solve using the Riccati equation on MATLAB. This is a sample example I got from a book Optimal Control Donald Kirk. I don't understand how they derived three separate differential equations from the Riccati equation:

The problem goes as followed:

Consider the system https://www.physicsforums.com/attachments/upload_2016-11-26_20-25-28-png.109458/
https://www.physicsforums.com/attachments/upload_2016-11-26_20-26-3-png.109459/

How did they go from a single equation K to three separate equations. I keep looking for resources but many other examples seem to skip this part. Thank for any help or input.

Homework Equations

https://www.physicsforums.com/attachments/upload_2016-11-26_20-26-46-png.109460/

The Attempt at a Solution

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[JavierOlivares]

https://www.physicsforums.com/attachments/upload_2016-11-26_20-39-19-png.109463/

I found a paper on this online that gives somewhat of an example of this problem.
https://www.physicsforums.com/attachments/upload_2016-11-26_20-40-26-png.109465/
https://www.physicsforums.com/attachments/upload_2016-11-26_20-41-4-png.109467/
It seems slightly different though. Sorry for any inconvenience.

 

[Ray Vickson]

Homework Statement

I was wondering if I can get some help on a Linear Regulator Problem for an Optimal Control Problem. Given a state equation and performance measure I am trying to solve using the Riccati equation on MATLAB. This is a sample example I got from a book Optimal Control Donald Kirk. I don't understand how they derived three separate differential equations from the Riccati equation:

The problem goes as followed:

Consider the system https://www.physicsforums.com/attachments/upload_2016-11-26_20-25-28-png.109458/
https://www.physicsforums.com/attachments/upload_2016-11-26_20-26-3-png.109459/

How did they go from a single equation K to three separate equations. I keep looking for resources but many other examples seem to skip this part. Thank for any help or input.

Homework Equations

https://www.physicsforums.com/attachments/upload_2016-11-26_20-26-46-png.109460/

The Attempt at a Solution

[/B]
View attachment 109470
View attachment 109471

The differential equations for $\mathbf{K}$ and its transpose $\mathbf{K}^T$ are the same; and (as your attachment in post #2 states), $\mathbf{K}(t_f) = \mathbf{S}$, where $\mathbf{S}$ is a symmetric matrix. Therefore, the solution $\mathbf{K}(t)$ is a symmetric matrix as well. Now just write the Ricatti equation for the symmetric matrix $\mathbf{K}$ in terms of its components:
$$\mathbf{K} = \pmatrix{K_{11}&K_{12}\\K_{12} & K_{22}} $$

 

[JavierOlivares]

I'm still a little confused. I understand that the matrix is symmetric. I just don't understand how they have it equal on the LHS a row of three 3x1 differential equations when K seems to be a 2x2. That's where I'm confused. I'm thinking this is some linear algebra property that's going over my head. I don't know if the notes I provided actually answer my question. Thanks for the response

 

[Ray Vickson]

I'm still a little confused. I understand that the matrix is symmetric. I just don't understand how they have it equal on the LHS a row of three 3x1 differential equations when K seems to be a 2x2. That's where I'm confused. I'm thinking this is some linear algebra property that's going over my head. I don't know if the notes I provided actually answer my question. Thanks for the response

View attachment 109555

I have not checked the picture: it is too messy and unstructured. However, both sides of your differential equation are 2x2 matrices, so you get 4 coupled differential equations. Since the matrix is symmetric, only three of the equations are different

 

[JavierOlivares]

I think I understand now. I was just confused on multiplying the X Matrix by another 2x2 Matrix. I was thinking the equations would combine X11 + X12 as in the case of a 2x2 and 2x1 but it makes sense now. Thanks.